Using the Very Large Array (VLA) with an angular resolution of 3", we have detected the hydrogen recombination line H92α from the galaxies Arp 220, M83, and NGC 2146. The line emission arises from the nuclear regions with a line-to-continuum ratio of 1% or less. In order to fit both the observed H92α line and continuum data in the nuclear regions, we have considered two types of models. First, we utilize a model with a collection of H II regions. A large number of compact H II regions are required in this model. With electron temperatures in the range 5 x 10^3^ - 1 x 10^4^ K and a range of electron densities, this model can account for both the line intensity and the continuum spectrum. In most cases, the H92α line is dominated by internal stimulated emission due to free-free continuum arising within the H II regions. In a low-density case (n_e_ = 50 cm^-3^) for Arp 220, about half the line emission comes from external stimulated emission due to the background nonthermal source. Typical rates of ionizing photons predicted from these models are ~5 x 10^52^ s^-1^ for M83, ~4 x 10^53^ s^-1^ for NGC 2146, and 5 x 10^54^ s^-1^ for Arp 220. We infer that 10^5^ O5 stars are required in Arp 220, which is an order of magnitude greater than in NGC 2146 and 2 orders of magnitude greater than in M83. Alternatively, several uniform slab models with T_e_ >= 5 x 10^3^ and n_e_ in the range of 50 - 1 x 10^4^ cm^-3^ appear to fit both the H92α line and continuum data of Arp 220 and M83. In the low-density models, stimulated emission by the background nonthermal radiation appears to be dominant at low frequencies, and the lines at higher frequencies arise primarily from spontaneous emission. The uniform slab model requires a higher ionizing photon rate than the H II region model. No slab models with reasonable T_e_ can fit the data observed in M83 and NGC 2146. Combining previous published data with these new observations, a sample of 13 galaxies has been observed for radio recombination lines (RRLs) with the VLA. Nine out of the 13 galaxies have been detected in the H92α line. While the H92α line luminosity appears to be correlated with the Brα line luminosity, we find that nearly all the RRL galaxies show a significant excess in H92α line compared to the expected LTE value. The excess in the H92α line flux suggests that non-LTE effects are important for the H92α line in these starburst nuclei. A strong correlation between H92α and the molecular lines of HCN/HCO^+^ is also found, indicating that the RRL emitters may be spatially associated with the dense molecular cores. The inferred high electron density also suggests an intimate relation between the RRLs and the dense molecular medium in these galaxies.